Born-Oppenheimer-breakdown (BOB) functions fitted to the spectra of alkali-metal dimers are 'effective' corrections that mix the true adiabatic diagonal Born-Oppenheimer correction (DBOC) with nonadiabatic terms, and for Li2_2 the phase-integral shift δvD\delta v_{\mathrm{D}} derived from ab initio DBOC alone falls short of the value inferred from Feshbach-resonance and near-threshold analyses by a factor of about five. The problem is to give a physically decomposed account of beyond-mass-scaling isotope shifts in Li2_2, LiK and LiRb — separating the true adiabatic DBOC from nonadiabatic contributions, using independent constraints that overcome Watson's identifiability theorem, with the theoretically required long-range behavior — such that a single model simultaneously reproduces spectroscopic line positions and near-threshold observables (binding energies, scattering lengths, Feshbach-resonance positions) for multiple isotopologues and the Li2_2 δvD\delta v_{\mathrm{D}} discrepancy is attributed to identified physical terms or to a demonstrated deficiency of the ab initio functions or of the empirical functional forms. The problem is open: the most recent Li2_2 analysis (2026) still absorbs near-threshold Born-Oppenheimer corrections into phenomenological inner-wall shift parameters, no decomposition via dipole moments, g-factors or combined data has been published for any alkali dimer, and the needed LiK/LiRb near-threshold data have not been measured. Reliable resolution matters because conventional mass- and field-shift calculations gate isotope-shift searches for new short-range forces in weakly bound molecules.

Background

In the Born-Oppenheimer (BO) approximation the electronic Schrödinger equation is solved at fixed nuclear positions, producing a potential energy curve V(R)V(R) that is by construction identical for all isotopologues of a molecule; isotope dependence enters the nuclear motion only through the reduced mass μ\mu ('BO mass scaling'). Deviations from this picture are called Born-Oppenheimer-breakdown (BOB) effects and have two distinct physical origins. First, the adiabatic correction, or diagonal Born-Oppenheimer correction (DBOC): the expectation value of the nuclear kinetic-energy operator over the electronic wavefunction, which shifts the potential itself by an isotope-dependent mass-shift function δVad(R)\delta V^{\mathrm{ad}}(R) proportional to inverse nuclear masses. Second, nonadiabatic corrections: because the electrons do not follow the nuclei perfectly, the effective vibrational and rotational reduced masses become functions of internuclear distance, μvib(R)\mu_{\mathrm{vib}}(R) and μrot(R)\mu_{\mathrm{rot}}(R), rather than constants. For heavy isotopologues there is in addition the isotope field shift: nuclei of different isotopes have different charge radii, and the resulting change in electron density at the nuclei (the 'contact density') shifts the potential by an amount proportional to the mean-square charge-radius difference.

Empirically, BOB effects are extracted by direct potential fits: an analytic potential plus small isotope-dependent correction functions ('BOB functions') is least-squares fitted to line positions of several isotopologues simultaneously. This has been done for Li2_2 (Coxon and Melville 2006; Le Roy et al. 2009), KLi (Tiemann et al. 2009) and LiRb (Ivanova et al. 2011), which were the only alkali-metal dimers with published ground-state BOB functions when Lutz and Hutson (2016) revisited the problem. A fundamental obstruction was established by Watson (1980, 2004): from transition frequencies alone one can determine only an 'effective' adiabatic correction that absorbs the nonadiabatic contribution to μvib\mu_{\mathrm{vib}}; the true DBOC and the nonadiabatic terms cannot be separated. The effective-Hamiltonian formulation of Herman and Ogilvie (1998) does permit a separation if an independent observable — the molecular dipole-moment function or the rotational g-factor (the rotational magnetic moment, which measures how the electrons co-rotate with the nuclei) — is supplied. This route has been demonstrated for light hydrides and CO, but has never been applied to an alkali-metal dimer.

Lutz and Hutson (2016) computed ab initio adiabatic mass-shift functions for the alkali dimers at the coupled-cluster level (a standard high-accuracy electron-correlation method of quantum chemistry) and compared them with the empirical BOB functions. They found reasonable agreement in magnitude near equilibrium for LiK and LiRb but qualitatively different shapes, plausibly because the empirical functions are effective (adiabatic plus nonadiabatic) corrections. They also connected BOB effects to ultracold physics, where the observables are near-threshold quantities. For a potential whose long range is the van der Waals tail −C6R−6-C_6 R^{-6}, the s-wave scattering length aa — the parameter governing low-energy collisions of ultracold atoms — is fixed semiclassically by the accumulated phase integral Φ\Phi at threshold, equivalently by a non-integer vibrational quantum number at dissociation vDv_{\mathrm{D}} whose fractional part is set by that phase. Magnetic Feshbach resonances (tunable avoided crossings between molecular bound states and scattering channels in a magnetic field) probe the same phase with high precision. The isotope shift δvD\delta v_{\mathrm{D}} of this threshold quantum number is therefore a model-aware measure of deviations from BO mass scaling near dissociation, and it is a delicate near-cancellation of short-range and long-range contributions.

For Li2_2, the coupled-channels analysis of Feshbach resonances and near-threshold bound states by Julienne and Hutson (2014) requires a non-BO contribution δvD=+9.4×10−4\delta v_{\mathrm{D}} = +9.4\times10^{-4} for the substitution 6^{6}Li2← 7_2\leftarrow\,^{7}Li2_2 (correspondingly +4.7×10−4+4.7\times10^{-4} for 6^{6}Li7^{7}Li← 7\leftarrow\,^{7}Li7^{7}Li), while the ab initio DBOC-derived phase integral of Lutz and Hutson accounts for only about one fifth of it. Julienne and Hutson also showed the converse of Watson's theorem from the scattering side: the deviation of scattering lengths from mass scaling constrains only the overall magnitude of the non-BO terms, not their RR-dependence. In addition, beyond the Le Roy radius (the distance outside which the interaction is a pure multipole expansion, where BOB theory requires δVad(R)/V(R)→\delta V^{\mathrm{ad}}(R)/V(R)\to const, i.e. δVad(R)∝R−6\delta V^{\mathrm{ad}}(R)\propto R^{-6}), the empirical BOB functions for LiK and LiRb do not converge to the expected behavior — their semiclassical integrands remain nearly constant, so that the corresponding phase integrals show no sign of converging by R=20R=20 bohr — while for Li2_2 the qualitative disagreement between the ab initio and empirical functions is at short range. Finally, the driving application: proposals exist to use isotope shifts of weakly bound Yb2_2 and Sr2_2 levels (measured by photoassociation spectroscopy (in which a laser binds pairs of colliding atoms into a molecule, so that the excitation spectrum probes weakly bound levels near dissociation) or by molecular lattice clocks (precision clocks based on weakly bound molecular transitions of atoms trapped in a standing-wave light field)) to constrain hypothetical short-range 'gravitylike' forces or new scalar bosons; before any anomaly is attributed to new physics, the conventional mass and field shifts must be computed reliably, which for heavy species such as Yb2_2 requires convergent four-component relativistic calculations of contact densities that in 2016 were only order-of-magnitude reliable.

Problem Statement

For the ground electronic states of the alkali-metal dimers Li2_2, LiK and LiRb — the only alkali-metal dimers with empirically fitted Born-Oppenheimer-breakdown functions in the spectroscopic literature at the time of Lutz and Hutson (2016), and workhorses of ultracold physics — construct a physically decomposed account of isotope shifts beyond Born-Oppenheimer mass scaling that resolves three linked discrepancies left open by Lutz and Hutson (2016):

  1. Decomposition: determine, for at least one of these dimers for which near-threshold data exist (currently Li2_2), the true adiabatic correction δVad(R)\delta V^{\mathrm{ad}}(R) (the DBOC) and the nonadiabatic correction (as distance-dependent vibrational and rotational reduced masses or an equivalent operator) separately, rather than only their merged effective adiabatic combination, overcoming Watson's theorem that line positions alone cannot separate them — using independent constraints such as molecular dipole-moment functions, rotational g-factors, or near-threshold/Feshbach data excluded from the spectroscopic fit.

  2. Reconciliation of the threshold phase integral: identify the physical origin of the factor-of-about-5 gap in Li2_2 between the shift δvD\delta v_{\mathrm{D}} inferred from Feshbach-resonance and near-threshold analyses (+9.4×10−4+9.4\times10^{-4} for 6^{6}Li2← 7_2\leftarrow\,^{7}Li2_2; +4.7×10−4+4.7\times10^{-4} for 6^{6}Li7^{7}Li← 7\leftarrow\,^{7}Li7^{7}Li) and the value obtained from the ab initio DBOC alone — establishing whether the missing contribution is nonadiabatic (and computing it), or reflects basis-set or method deficiencies in the ab initio functions, or flawed functional forms in the empirical fits.

  3. Asymptotic behavior: obtain fitted BOB functions that display the theoretically required long-range form δVad(R)/V(R)→\delta V^{\mathrm{ad}}(R)/V(R)\to const (hence δVad(R)∝R−6\delta V^{\mathrm{ad}}(R)\propto R^{-6}) beyond the Le Roy radius, replacing empirical forms whose semiclassical integrands show no convergence (the failure established by Lutz and Hutson (2016) for LiK and LiRb); this refit of the LiK and LiRb empirical forms is an independent sub-goal and is not required by the single-model completion criterion, which is pegged to the dimer currently having near-threshold data (Li2_2).

The problem is complete when a single model with separated adiabatic and nonadiabatic components — the separation being established through constraints independent of the line positions used to fix the effective correction, as in goal 1 (for example dipole-moment functions, rotational g-factors, near-threshold/Feshbach data excluded from the spectroscopic fit, or independent nonadiabatic theory input), not by defining one component as the residual of the other — simultaneously reproduces, within experimental uncertainties, high-resolution spectroscopic line positions of deeply bound levels and near-threshold observables (binding energies, scattering lengths, Feshbach-resonance positions) for at least two isotopologues of at least one of these dimers for which such near-threshold data exist (currently Li2_2), exhibits, for that dimer, the correct asymptotic behavior, and the Li2_2 δvD\delta v_{\mathrm{D}} discrepancy is quantitatively attributed to identified physical terms or to a demonstrated deficiency of the ab initio functions or of the empirical functional forms. The following enabling subtasks would serve the same objective but are not part of the completion criterion — new near-threshold spectra and Feshbach-resonance positions for both Li isotopologues of LiK and LiRb, and convergent four-component relativistic calculations of contact densities (field shifts) for heavy systems such as Yb2_2 — which would allow isotope-dependent anomalies at the level relevant to fifth-force searches to be attributed to new physics rather than conventional mass- and field-shift effects.

References

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